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arXiv · 2607.09163

Beyond F5 and GVW: The Proper-Cover Algorithm for Fast Ideal Basis Computation

Sheng-Ming Ma·Yi Liu·Zheng-Lin Jiao
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Abstract

Gr\"obner basis computation incurs heavy computational overhead, especially under lexicographic order. F5 and its GVW variant dominate efficient field-based Gr\"obner basis solving. The proper basis algorithm offers a parameterized ideal computation framework without leveraging modern signature-based optimizations. This work presents the Proper-Cover algorithm for zero-dimensional polynomial ideals by combining GVW's cover optimization over signature with the proper basis theory. We generalize signature, cover, POT ordering, reduction and S-pair concepts to parameterized coefficients, design a two-phase algorithm with compatible factor construction and hungry refinement, and rigorously prove termination and output correctness. Accordingly, we propose a new framework for the efficient computation of polynomial ideal bases. Benchmark results show that Proper-Cover surpasses F5 under all monomial orderings and delivers clear speedups over GVW for lexicographic (plex) order.

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Sheng-Ming Ma, Yi Liu, Zheng-Lin Jiao. 2026-07-10. Beyond F5 and GVW: The Proper-Cover Algorithm for Fast Ideal Basis Computation. https://arxiv.org/abs/2607.09163

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